What is the integral of 12cos(4x) ?

The integral of 12cos(4x) is 3sin(4x)+C

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integral of 12cos(4x)

\int 12 cos (4 x) d x

3 sin (4 x) + C

Solution steps

\int 12 cos (4 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 12 \cdot \int cos (4 x) d x

Apply u-substitution

= 12 \cdot \int cos (u) \frac{1}{4} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 12 \cdot \frac{1}{4} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= 12 \cdot \frac{1}{4} sin (u)

Substitute back

u = 4 x

= 12 \cdot \frac{1}{4} sin (4 x)

Simplify

12 \cdot \frac{1}{4} sin (4 x) : 3 sin (4 x)

= 3 sin (4 x)

Add a constant to the solution

= 3 sin (4 x) + C